A Fictitious Time Integration Method for Solving$m$-Point Boundary Value Problems

Abstract

We propose a new numerical method for solving the boundary value problems of ordinary differential equations (ODEs) under multipoint boundary conditions specified at t = Ti; i = 1;:::;m, where T1 < ::: < Tm. The finite dif- ference scheme is used to approximate the ODEs, which together with the m-point boundary conditions constitute a system of nonlinear algebraic equations (NAEs). Then a Fictitious Time Integration Method (FTIM) is used to solve these NAEs. Numerical examples confirm that the new approach is highly accurate and efficient with a fast convergence. The FTIM can also be used to find the periods of nonlin- ear ODEs system and its corresponding periodic solutions, as the van der Pol and Duffing equations are investigated here. The numerical examples also include a vibration problem of the Euler-Bernoulli beam under three-point boundary condi- tions. The present method has a number advantages of easy implementation, easily to treat nonlinear multipoint boundary value problems, and easily to extend to a higher-dimensional first-order ODEs.